What is the integral of (sin(7e^{-5x}))/(e^{5x)} ?

The integral of (sin(7e^{-5x}))/(e^{5x)} is 1/35 cos(7e^{-5x})+C

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integral of (sin(7e^{-5x}))/(e^{5x)}

\int \frac{sin ( 7 e ^{- 5 x} )}{e ^{5 x}} d x

\frac{1}{35} cos (7 e^{- 5 x}) + C

Solution steps

\int \frac{sin ( 7 e ^{- 5 x} )}{e ^{5 x}} d x

Apply u-substitution

: \int - \frac{sin ( u )}{35} d u

= \int - \frac{sin ( u )}{35} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \frac{1}{35} \cdot \int sin (u) d u

Use the common integral:

\int sin (u) d u = - cos (u)

= - \frac{1}{35} (- cos (u))

Substitute back

u = 7 e^{- 5 x}

= - \frac{1}{35} (- cos (7 e^{- 5 x}))

Simplify

= \frac{1}{35} cos (7 e^{- 5 x})

Add a constant to the solution

= \frac{1}{35} cos (7 e^{- 5 x}) + C

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What is the integral of (sin(7e^{-5x}))/(e^{5x)} ?
The integral of (sin(7e^{-5x}))/(e^{5x)} is 1/35 cos(7e^{-5x})+C